Normalized defining polynomial
\( x^{22} - 19 x^{20} + 110 x^{18} - 228 x^{16} - 81 x^{14} + 1073 x^{12} - 1656 x^{10} + 934 x^{8} + \cdots - 1 \)
Invariants
Degree: | $22$ | sage: K.degree()
gp: poldegree(K.pol)
magma: Degree(K);
oscar: degree(K)
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Signature: | $[18, 2]$ | sage: K.signature()
gp: K.sign
magma: Signature(K);
oscar: signature(K)
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Discriminant: | \(56501459388151144478039723653407440896\) \(\medspace = 2^{22}\cdot 1297^{10}\) | sage: K.disc()
gp: K.disc
magma: OK := Integers(K); Discriminant(OK);
oscar: OK = ring_of_integers(K); discriminant(OK)
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Root discriminant: | \(52.00\) | sage: (K.disc().abs())^(1./K.degree())
gp: abs(K.disc)^(1/poldegree(K.pol))
magma: Abs(Discriminant(OK))^(1/Degree(K));
oscar: (1.0 * dK)^(1/degree(K))
| |
Galois root discriminant: | not computed | ||
Ramified primes: | \(2\), \(1297\) | sage: K.disc().support()
gp: factor(abs(K.disc))[,1]~
magma: PrimeDivisors(Discriminant(OK));
oscar: prime_divisors(discriminant((OK)))
| |
Discriminant root field: | \(\Q\) | ||
$\card{ \Aut(K/\Q) }$: | $2$ | sage: K.automorphisms()
magma: Automorphisms(K);
oscar: automorphisms(K)
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This field is not Galois over $\Q$. | |||
This is not a CM field. |
Integral basis (with respect to field generator \(a\))
$1$, $a$, $a^{2}$, $a^{3}$, $a^{4}$, $a^{5}$, $a^{6}$, $a^{7}$, $a^{8}$, $a^{9}$, $a^{10}$, $a^{11}$, $a^{12}$, $a^{13}$, $a^{14}$, $a^{15}$, $a^{16}$, $a^{17}$, $\frac{1}{5}a^{18}-\frac{1}{5}a^{16}-\frac{1}{5}a^{14}+\frac{2}{5}a^{12}-\frac{2}{5}a^{10}+\frac{1}{5}a^{8}-\frac{2}{5}a^{6}-\frac{2}{5}$, $\frac{1}{5}a^{19}-\frac{1}{5}a^{17}-\frac{1}{5}a^{15}+\frac{2}{5}a^{13}-\frac{2}{5}a^{11}+\frac{1}{5}a^{9}-\frac{2}{5}a^{7}-\frac{2}{5}a$, $\frac{1}{145}a^{20}+\frac{4}{145}a^{18}-\frac{1}{145}a^{16}-\frac{48}{145}a^{14}+\frac{33}{145}a^{12}-\frac{24}{145}a^{10}-\frac{62}{145}a^{8}+\frac{6}{29}a^{6}+\frac{2}{29}a^{4}+\frac{18}{145}a^{2}-\frac{1}{29}$, $\frac{1}{145}a^{21}+\frac{4}{145}a^{19}-\frac{1}{145}a^{17}-\frac{48}{145}a^{15}+\frac{33}{145}a^{13}-\frac{24}{145}a^{11}-\frac{62}{145}a^{9}+\frac{6}{29}a^{7}+\frac{2}{29}a^{5}+\frac{18}{145}a^{3}-\frac{1}{29}a$
Monogenic: | Not computed | |
Index: | $1$ | |
Inessential primes: | None |
Class group and class number
Trivial group, which has order $1$ (assuming GRH)
Unit group
Rank: | $19$ | sage: UK.rank()
gp: K.fu
magma: UnitRank(K);
oscar: rank(UK)
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Torsion generator: | \( -1 \) (order $2$) | sage: UK.torsion_generator()
gp: K.tu[2]
magma: K!f(TU.1) where TU,f is TorsionUnitGroup(K);
oscar: torsion_units_generator(OK)
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Fundamental units: | $a$, $\frac{697}{29}a^{20}-\frac{61199}{145}a^{18}+\frac{295969}{145}a^{16}-\frac{380691}{145}a^{14}-\frac{774773}{145}a^{12}+\frac{2567018}{145}a^{10}-\frac{2252624}{145}a^{8}+\frac{467688}{145}a^{6}+\frac{26226}{29}a^{4}-\frac{10161}{29}a^{2}+\frac{3948}{145}$, $\frac{732}{145}a^{20}-\frac{13051}{145}a^{18}+\frac{65562}{145}a^{16}-\frac{95717}{145}a^{14}-\frac{145262}{145}a^{12}+\frac{117225}{29}a^{10}-\frac{604678}{145}a^{8}+\frac{195728}{145}a^{6}+\frac{4103}{29}a^{4}-\frac{20319}{145}a^{2}+\frac{2198}{145}$, $\frac{2493}{145}a^{21}-\frac{45853}{145}a^{19}+\frac{247487}{145}a^{17}-\frac{437069}{145}a^{15}-\frac{380716}{145}a^{13}+\frac{2354853}{145}a^{11}-\frac{2973656}{145}a^{9}+\frac{246349}{29}a^{7}+\frac{21893}{29}a^{5}-\frac{108246}{145}a^{3}+\frac{2147}{29}a$, $\frac{266}{145}a^{21}-\frac{8013}{145}a^{19}+\frac{80151}{145}a^{17}-\frac{293546}{145}a^{15}+\frac{217694}{145}a^{13}+\frac{205818}{29}a^{11}-\frac{2352384}{145}a^{9}+\frac{1474394}{145}a^{7}+\frac{5607}{29}a^{5}-\frac{119767}{145}a^{3}+\frac{13054}{145}a$, $\frac{5116}{145}a^{21}-\frac{89649}{145}a^{19}+\frac{431102}{145}a^{17}-\frac{108465}{29}a^{15}-\frac{1158763}{145}a^{13}+\frac{3723602}{145}a^{11}-\frac{632169}{29}a^{9}+\frac{567136}{145}a^{7}+\frac{41378}{29}a^{5}-\frac{65962}{145}a^{3}+\frac{4406}{145}a$, $\frac{64}{5}a^{21}-\frac{1121}{5}a^{19}+\frac{5388}{5}a^{17}-1361a^{15}-\frac{14262}{5}a^{13}+\frac{46178}{5}a^{11}-8027a^{9}+\frac{9174}{5}a^{7}+262a^{5}-\frac{1073}{5}a^{3}+\frac{134}{5}a$, $\frac{2426}{145}a^{21}-\frac{41481}{145}a^{19}+\frac{186654}{145}a^{17}-\frac{175318}{145}a^{15}-\frac{634647}{145}a^{13}+\frac{1503426}{145}a^{11}-\frac{821327}{145}a^{9}-\frac{31467}{29}a^{7}+\frac{13842}{29}a^{5}+\frac{5098}{145}a^{3}-\frac{338}{29}a$, $\frac{732}{145}a^{20}-\frac{13051}{145}a^{18}+\frac{65562}{145}a^{16}-\frac{95717}{145}a^{14}-\frac{145262}{145}a^{12}+\frac{117225}{29}a^{10}-\frac{604678}{145}a^{8}+\frac{195728}{145}a^{6}+\frac{4103}{29}a^{4}-\frac{20319}{145}a^{2}+\frac{2053}{145}$, $\frac{732}{145}a^{20}-\frac{13051}{145}a^{18}+\frac{65562}{145}a^{16}-\frac{95717}{145}a^{14}-\frac{145262}{145}a^{12}+\frac{117225}{29}a^{10}-\frac{604678}{145}a^{8}+\frac{195728}{145}a^{6}+\frac{4103}{29}a^{4}-\frac{20464}{145}a^{2}+\frac{2343}{145}$, $\frac{2623}{145}a^{21}-\frac{4812}{145}a^{20}-\frac{43796}{145}a^{19}+\frac{84398}{145}a^{18}+\frac{36723}{29}a^{17}-\frac{406814}{145}a^{16}-\frac{105256}{145}a^{15}+\frac{103302}{29}a^{14}-\frac{778047}{145}a^{13}+\frac{1081041}{145}a^{12}+\frac{1368749}{145}a^{11}-\frac{3516269}{145}a^{10}-\frac{187189}{145}a^{9}+\frac{605762}{29}a^{8}-\frac{664609}{145}a^{7}-\frac{595252}{145}a^{6}+\frac{19485}{29}a^{5}-\frac{34796}{29}a^{4}+\frac{42284}{145}a^{3}+\frac{68824}{145}a^{2}-\frac{6329}{145}a-\frac{5607}{145}$, $\frac{6257}{145}a^{21}-\frac{10853}{145}a^{20}-\frac{108546}{145}a^{19}+\frac{189313}{145}a^{18}+\frac{508377}{145}a^{17}-\frac{899602}{145}a^{16}-\frac{576967}{145}a^{15}+\frac{1082094}{145}a^{14}-\frac{1504577}{145}a^{13}+\frac{2526771}{145}a^{12}+\frac{854681}{29}a^{11}-\frac{7672768}{145}a^{10}-\frac{3158043}{145}a^{9}+\frac{6143301}{145}a^{8}+\frac{261863}{145}a^{7}-\frac{175782}{29}a^{6}+\frac{43428}{29}a^{5}-\frac{78401}{29}a^{4}-\frac{48614}{145}a^{3}+\frac{117121}{145}a^{2}+\frac{3138}{145}a-\frac{1646}{29}$, $\frac{2077}{145}a^{21}+\frac{416}{145}a^{20}-\frac{38701}{145}a^{19}-\frac{7152}{145}a^{18}+\frac{214727}{145}a^{17}+\frac{6552}{29}a^{16}-\frac{402572}{145}a^{15}-\frac{34497}{145}a^{14}-\frac{280677}{145}a^{13}-\frac{100039}{145}a^{12}+\frac{418140}{29}a^{11}+\frac{264153}{145}a^{10}-\frac{2786073}{145}a^{9}-\frac{187583}{145}a^{8}+\frac{1202793}{145}a^{7}+\frac{28952}{145}a^{6}+\frac{22453}{29}a^{5}-\frac{560}{29}a^{4}-\frac{102249}{145}a^{3}-\frac{5997}{145}a^{2}+\frac{9828}{145}a+\frac{907}{145}$, $\frac{931}{145}a^{21}-\frac{143}{29}a^{20}-\frac{17243}{145}a^{19}+\frac{13612}{145}a^{18}+\frac{94566}{145}a^{17}-\frac{78977}{145}a^{16}-\frac{174086}{145}a^{15}+\frac{162703}{145}a^{14}-\frac{126631}{145}a^{13}+\frac{67784}{145}a^{12}+\frac{181253}{29}a^{11}-\frac{789359}{145}a^{10}-\frac{1210269}{145}a^{9}+\frac{1166717}{145}a^{8}+\frac{549524}{145}a^{7}-\frac{571319}{145}a^{6}+\frac{5371}{29}a^{5}-\frac{5548}{29}a^{4}-\frac{48782}{145}a^{3}+\frac{9693}{29}a^{2}+\frac{5959}{145}a-\frac{5299}{145}$, $\frac{1291}{145}a^{21}-\frac{5116}{145}a^{20}-\frac{24213}{145}a^{19}+\frac{89649}{145}a^{18}+\frac{136111}{145}a^{17}-\frac{431102}{145}a^{16}-\frac{261546}{145}a^{15}+\frac{108465}{29}a^{14}-\frac{165791}{145}a^{13}+\frac{1158763}{145}a^{12}+\frac{268004}{29}a^{11}-\frac{3723602}{145}a^{10}-\frac{1815924}{145}a^{9}+\frac{632169}{29}a^{8}+\frac{788699}{145}a^{7}-\frac{567136}{145}a^{6}+\frac{15023}{29}a^{5}-\frac{41378}{29}a^{4}-\frac{63182}{145}a^{3}+\frac{65962}{145}a^{2}+\frac{6189}{145}a-\frac{4551}{145}$, $\frac{6374}{145}a^{21}-\frac{279}{145}a^{20}-\frac{113008}{145}a^{19}+\frac{4249}{145}a^{18}+\frac{111947}{29}a^{17}-\frac{12481}{145}a^{16}-\frac{779493}{145}a^{15}-\frac{21118}{145}a^{14}-\frac{1335126}{145}a^{13}+\frac{115783}{145}a^{12}+\frac{4964857}{145}a^{11}-\frac{41154}{145}a^{10}-\frac{4792197}{145}a^{9}-\frac{245587}{145}a^{8}+\frac{1277473}{145}a^{7}+\frac{46988}{29}a^{6}+\frac{51318}{29}a^{5}+\frac{196}{29}a^{4}-\frac{128288}{145}a^{3}-\frac{16767}{145}a^{2}+\frac{10818}{145}a+\frac{337}{29}$, $\frac{4024}{145}a^{21}+\frac{204}{145}a^{20}-\frac{14407}{29}a^{19}-\frac{2983}{145}a^{18}+\frac{365407}{145}a^{17}+\frac{1415}{29}a^{16}-\frac{548866}{145}a^{15}+\frac{24022}{145}a^{14}-\frac{155666}{29}a^{13}-\frac{91346}{145}a^{12}+\frac{3306371}{145}a^{11}+\frac{2267}{145}a^{10}-\frac{3508334}{145}a^{9}+\frac{241508}{145}a^{8}+\frac{1142622}{145}a^{7}-\frac{216687}{145}a^{6}+\frac{32205}{29}a^{5}+\frac{1742}{29}a^{4}-\frac{109253}{145}a^{3}+\frac{17012}{145}a^{2}+\frac{10272}{145}a-\frac{2122}{145}$, $\frac{266}{145}a^{21}-\frac{1258}{145}a^{20}-\frac{8013}{145}a^{19}+\frac{23359}{145}a^{18}+\frac{80151}{145}a^{17}-\frac{128633}{145}a^{16}-\frac{293546}{145}a^{15}+\frac{237168}{145}a^{14}+\frac{217694}{145}a^{13}+\frac{176363}{145}a^{12}+\frac{205818}{29}a^{11}-\frac{248251}{29}a^{10}-\frac{2352384}{145}a^{9}+\frac{1631352}{145}a^{8}+\frac{1474394}{145}a^{7}-\frac{710337}{145}a^{6}+\frac{5607}{29}a^{5}-\frac{9940}{29}a^{4}-\frac{119767}{145}a^{3}+\frac{62326}{145}a^{2}+\frac{13054}{145}a-\frac{6412}{145}$, $\frac{1291}{145}a^{21}+\frac{4689}{145}a^{20}-\frac{24213}{145}a^{19}-\frac{80714}{145}a^{18}+\frac{136111}{145}a^{17}+\frac{370136}{145}a^{16}-\frac{261546}{145}a^{15}-\frac{382832}{145}a^{14}-\frac{165791}{145}a^{13}-\frac{1177523}{145}a^{12}+\frac{268004}{29}a^{11}+\frac{3041069}{145}a^{10}-\frac{1815924}{145}a^{9}-\frac{1960683}{145}a^{8}+\frac{788699}{145}a^{7}-\frac{10146}{29}a^{6}+\frac{15023}{29}a^{5}+\frac{28257}{29}a^{4}-\frac{63182}{145}a^{3}-\frac{16373}{145}a^{2}+\frac{6189}{145}a+\frac{67}{29}$ (assuming GRH) | sage: UK.fundamental_units()
gp: K.fu
magma: [K|fUK(g): g in Generators(UK)];
oscar: [K(fUK(a)) for a in gens(UK)]
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Regulator: | \( 244248936396 \) (assuming GRH) | sage: K.regulator()
gp: K.reg
magma: Regulator(K);
oscar: regulator(K)
|
Class number formula
\[ \begin{aligned}\lim_{s\to 1} (s-1)\zeta_K(s) =\mathstrut & \frac{2^{r_1}\cdot (2\pi)^{r_2}\cdot R\cdot h}{w\cdot\sqrt{|D|}}\cr \approx\mathstrut &\frac{2^{18}\cdot(2\pi)^{2}\cdot 244248936396 \cdot 1}{2\cdot\sqrt{56501459388151144478039723653407440896}}\cr\approx \mathstrut & 0.168140568509173 \end{aligned}\] (assuming GRH)
Galois group
$C_2^{10}.D_{11}$ (as 22T30):
A solvable group of order 22528 |
The 100 conjugacy class representatives for $C_2^{10}.D_{11}$ |
Character table for $C_2^{10}.D_{11}$ |
Intermediate fields
11.11.3670285774226257.1 |
Fields in the database are given up to isomorphism. Isomorphic intermediate fields are shown with their multiplicities.
Sibling fields
Degree 22 siblings: | data not computed |
Degree 44 siblings: | data not computed |
Minimal sibling: | 22.10.73282392826432034388017521578469450842112.6 |
Frobenius cycle types
$p$ | $2$ | $3$ | $5$ | $7$ | $11$ | $13$ | $17$ | $19$ | $23$ | $29$ | $31$ | $37$ | $41$ | $43$ | $47$ | $53$ | $59$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
Cycle type | R | ${\href{/padicField/3.11.0.1}{11} }^{2}$ | ${\href{/padicField/5.4.0.1}{4} }^{5}{,}\,{\href{/padicField/5.2.0.1}{2} }$ | ${\href{/padicField/7.11.0.1}{11} }^{2}$ | ${\href{/padicField/11.4.0.1}{4} }^{2}{,}\,{\href{/padicField/11.2.0.1}{2} }^{6}{,}\,{\href{/padicField/11.1.0.1}{1} }^{2}$ | ${\href{/padicField/13.11.0.1}{11} }^{2}$ | ${\href{/padicField/17.4.0.1}{4} }^{3}{,}\,{\href{/padicField/17.2.0.1}{2} }^{5}$ | ${\href{/padicField/19.11.0.1}{11} }^{2}$ | ${\href{/padicField/23.11.0.1}{11} }^{2}$ | ${\href{/padicField/29.4.0.1}{4} }^{3}{,}\,{\href{/padicField/29.2.0.1}{2} }^{5}$ | ${\href{/padicField/31.4.0.1}{4} }{,}\,{\href{/padicField/31.2.0.1}{2} }^{9}$ | ${\href{/padicField/37.4.0.1}{4} }^{3}{,}\,{\href{/padicField/37.2.0.1}{2} }^{5}$ | ${\href{/padicField/41.4.0.1}{4} }^{3}{,}\,{\href{/padicField/41.2.0.1}{2} }^{5}$ | ${\href{/padicField/43.4.0.1}{4} }^{4}{,}\,{\href{/padicField/43.2.0.1}{2} }^{2}{,}\,{\href{/padicField/43.1.0.1}{1} }^{2}$ | ${\href{/padicField/47.11.0.1}{11} }^{2}$ | ${\href{/padicField/53.11.0.1}{11} }^{2}$ | ${\href{/padicField/59.4.0.1}{4} }^{3}{,}\,{\href{/padicField/59.2.0.1}{2} }^{5}$ |
In the table, R denotes a ramified prime. Cycle lengths which are repeated in a cycle type are indicated by exponents.
Local algebras for ramified primes
$p$ | Label | Polynomial | $e$ | $f$ | $c$ | Galois group | Slope content |
---|---|---|---|---|---|---|---|
\(2\) | Deg $22$ | $2$ | $11$ | $22$ | |||
\(1297\) | $\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ |
$\Q_{1297}$ | $x$ | $1$ | $1$ | $0$ | Trivial | $[\ ]$ | |
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $2$ | $2$ | $1$ | $1$ | $C_2$ | $[\ ]_{2}$ | ||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ | ||||
Deg $4$ | $2$ | $2$ | $2$ |